Effective behavior of heterogeneous media governed by strain gradient elasticity

TL;DR

The paper analyzes a 1D periodic heterogeneous medium governed by strain‑gradient elasticity to understand length‑scale effects across microstructure and macro scales. It shows that simple averaging fails to recover SG elasticity, while a nonlocal kernel with oscillatory behavior can describe the response; in a finite range of scales, a fractional strain‑gradient elasticity provides a tunable scaling between classical and SG regimes. A data‑driven approach using Fourier neural operators demonstrates accurate, discretization‑invariant learning of the nonlocal material response across scales. Together, these results offer a principled description of scale‑dependent behavior and a flexible framework for predicting nonlocal effects in heterogeneous media, with potential extensions to plasticity and complex rheologies.

Abstract

Various mechanical phenomena depend on the length scale, and these have inspired a variety of nonlocal and higher gradient continuum theories. Mechanistically, it is believed that the length scale dependence arises due to an interplay between the length scale of heterogeneities in the material, the length scale of the material being probed and the phenomenon under study. In this paper, we seek to understand this interplay in a simple setting by studying the overall behavior of a one-dimensional periodic medium governed by strain gradient elasticity at the microstructural scale. We find through numerical experiments that the overall behavior is not described by a strain gradient elasticity. In other words, strain gradient theories are not invariant under averaging at this scale. We also find that the overall behavior may be described by a kernel-based nonlocal elasticity theory, but the kernel is highly oscillatory with slow decay. So we seek alternate characterization. First, we limit our interest to a range of length scales, and show that the behavior is described well by fractional strain gradient elasticity. Consequently, one can obtain various scaling laws with exponent between zero (classical elasticity) and one (strain-gradient elasticity). Second, we take a data-driven approach, and show that we can describe the overall behavior over a range of scales using a Fourier neural operator.

Figures (7)

• Figure 1: Displacement (a) and strain (b) in a homogeneous hanging bar for various material length scales $\lambda$. All lengths are normalized with the length of bar $L$.
• Figure 2: Strain and effective elastic modulus of a heterogeneous bar with $E_1=1$ and $E_2=5$. (a) Strain in a two-piece bar ($\ell=1$) for various combinations of material lengths $\lambda_1, \lambda_2$. (b) Effective elastic modulus of a two-piece bar ($\ell=1$) for various combinations of material lengths. (c) Effective elastic modulus for $\ell=1, 1/10$ as a function of the material length when $\lambda_1=\lambda_2=\lambda$. (d) Effective elastic modulus as a function of $\ell$ for large and small material lengths ($\lambda_1=\lambda_2$). The upper red dashed line is the arithmetic mean 3 and lower black dashed line is the harmonic mean 5/3 of the elastic moduli. All lengths and normalized so that the length of the bar is 1.
• Figure 3: Response of a periodic heterogeneous bar with $E_1=1, E_2=5$ subjected to sinusoidal loading (a) Displacement and strain over four loading wavelengths with $k=1$ ($L=2\pi$), (b) Amplification $\hat{G}$ and (c) $\hat{G}k^2$ for various material length scales. All parts have the same combination and order of material lengths $\lambda$ as stated in the legend of (c), and all lengths and normalized so that the unit cell length $\ell=1$. We fix $E_1=1, E_2=5$
• Figure 4: The effective response of a heterogeneous bar. (a) The best fit effective elastic modulus $\bar{E}$ and effective material length $\bar{\lambda}$ as a function of the material length $\lambda_2$ with $\lambda_1=2$. (b) Comparison between the best effective strain gradient elasticity and actual response. (c) The kernels of nonlocal elasticity that describe the effective behavior a few combination of material lengths and (d) Empirical relationship between the exponent $\phi$ (cf. (\ref{['eq:phi']}) and scaled wavenumber $k\Lambda$. $E_1=1, E_2=5$.
• Figure 5: Learning the linear map. (a) Training and test error. (b) Comparison of the displacement field for four instances of test data. (c) Comparison of the displacement field for four instances of test data generated at a higher resolution than that used for training. (d) Comparison of the displacement field for four instances of test data generated at a lower resolution than that used for training. $E_1=1, E_2=5$.